Developing equation for specific wave

[infraray]

First off I apologize if this is in the wrong section, but I was not certain to where it belonged. If I am given a set of requirements for an e-m wave, i.e.: must contain all following: transverse, 3d, plane wave, propagation vector parallel to (-1,0,1), through plane defined by (1,0,1) and (0,1,0), etc (from memory). How would I begin to write an equation for this? I assuming I must start with say: Ae^ik[(alpha(x)+beta(y)+gamma(z))±wt] or some variant, but I am a bit weak when it comes to crafting equations. Perhaps I am lacking in my pedagogy, I just can't seem to see the big picture. Any help or recommendations would be greatly appreciated.

 

[Tide]

Your wave will have the form
$$\vec E = \vec E_0 e^{i \left( \vec k \cdot \vec x - \omega t \right)}$$
where $\vec k$ is in the direction of propagation and $\vec E _0$ is perpendicular to $\vec k$ and in the direction of polarization.

 

[M98Ranger]

Hello,

Thank you for reaching out with your question. Developing equations for specific waves can be a challenging task, but with some guidance, it can become easier. Let's break down the requirements given and see how we can incorporate them into an equation.

1. Transverse wave: A transverse wave is a type of wave where the displacement of the medium is perpendicular to the direction of propagation. This means that the wave will have a displacement in the x and z directions, while the propagation will be in the y direction.

2. 3D wave: A 3D wave means that the wave will have displacement in all three dimensions (x, y, z).

3. Plane wave: A plane wave is a type of wave where the wavefronts are flat and parallel, meaning the wave will have a constant amplitude and phase across the entire wavefront.

4. Propagation vector: The propagation vector, also known as the wave vector, represents the direction and magnitude of the wave's propagation. In this case, the vector is given as (-1, 0, 1), which means the wave is propagating in the -x direction, has no propagation in the y direction, and is propagating in the z direction.

5. Through plane defined by (1, 0, 1) and (0, 1, 0): This means that the wave must pass through the points (1, 0, 1) and (0, 1, 0). This can help us determine the direction of the wave's displacement.

Based on these requirements, we can start with the general equation for a transverse wave:

y(x, y, z, t) = A sin(kx ± ωt)

Here, A represents the amplitude of the wave, k represents the wave number, ω represents the angular frequency, and t represents time.

To incorporate the 3D aspect, we can add a second term for the displacement in the z direction:

y(x, y, z, t) = A sin(kx ± ωt) + B sin(kz ± ωt)

Next, we need to determine the values for A and B. Since the wave is transverse, the displacement must be perpendicular to the propagation. This means that A and B must be equal but opposite in direction. So, we can set A = -B.

Now, we can incorporate the propagation vector by using the dot product